How do you use the integral test to determine the convergence or divergence of #sum_(n=1)^(infty) 3/(n^(5/3))#?

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Answer 1 · 2018-02-04T10:25:38

The series converges

The series can be written as

#sum_(n=1)^oo3/(n^(5/3))=3sum_(n=1)^oo1/(n^(5/3))#

Let #f(x)=1/(x^(5/3#

This function #f(x)# is continuous, positive and decreasing on the Interval #(1,+oo)#, so we can apply the integral test

The integral

#int_1^(+oo)f(x)dx=int_1^(+oo)(1dx)/(x^(5/3)# converges as

#lim_(x->+oo)int_1^x(x^(-5/3)dx)=lim_(x->+oo)[-3/2x^(-2/3)]_1^x#

#=lim_(x->+oo)(3/2x^(-2/3))+3/2#

#=0+3/2#

#=3/2#

Therefore,

#sum_(n=1)^oo3/(n^(5/3))# converges to #=3*3/2=9/2#